Optical transmission of high speed data has led to the rapid expansion of the Internet. Modulation techniques such as quadrature phase shift keying (QPSK) and multilevel quadrature amplitude modulation (nQAM) have transformed communication from simple kilobaud-speed transmission of data one bit at a time to simultaneous transmission of multiple stream/multiple bit symbol data at terabit-speeds.
In a typical conventional optical data transmission system, a digital signal processor (DSP) converts the original digital data into high-speed signals. An optical modulator uses these high-speed signals to create a modulated analog light signal that carries the data over the optical fiber network. On the other end of the fiber network, an optical receiver detects the modulated analog light signal and a DSP extracts the digital information carried on the analog light signal.
FIG. 1 shows an example of a typical conventional polarization-multiplexed (PM) nQAM optical transmission system where transmitter 100 converts digital data 102 to modulated optical signal 114 which is received by receiver 130 and converted back to digital data 102. DSP 104 creates high-speed data signals 106XI, 106XQ, 106YI, and 106YQ from digital data 102; data signals 106XI, 106XQ, 106YI, and 106YQ represent the in-phase (I) and quadrature (Q) components for the X-polarization and Y-polarization signals. Data signals 106XI, 106XQ, 106YI, and 106YQ are sent (via radio frequency (RF) drivers, not shown) to Mach Zehnder modulator (MZM) 110 which imposes the data signals onto coherent carrier wave 112 emitted by laser diode 108 to produce modulated optical signal 114. Optical signal 114 travels from transmitter 100 through optical network 116 to receiver 130, where integrated coherent receiver (ICR) 122 which extracts data signals 126XI, 126XQ, 126YI, and 126YQ using local oscillator (LO) 128; data signals 126XI, 126XQ, 126YI, and 126YQ are a mix of data signals 106XI, 106XQ, 106YI, and 106YQ. ICR 122 sends data signals 126XI, 126XQ, 126YI, and 126YQ to DSP 124 (via analog-to-digital converters, not shown), which applies an inverse of an estimated transfer function to reproduce digital data 102.
The modulated optical signal 114 of the conventional system shown in FIG. 1 contains densely packed information. With the increased density of information comes a technical challenge of designing equipment that can efficiently and accurately transmit and extract the original data. While manufacturing techniques have vastly improved the quality of optical equipment, even slight imperfections in the optical and electrical components can distort data signals 106n and 126n as well as optical signal 114. The result is that version of digital data 102 that exits receiver 130 may differ from the version of digital data 102 that initially entered transmitter 100.
One way to correct for transmission impairments is to tighten the specifications of the optical and electrical components, however cost and availability make this approach impractical. A more practical alternative is to modify the transfer function used by the DSP so that it knows not only how to extract the original data from the sampled data set, but also how to correct for distortions made to the optical signal during modulation, transmission, and demodulation. While some distortions can be predicted in advance, most distortions are a function of the specific network configuration—the modulation and demodulation equipment, the fiber length and shape, the number and quality of connections, and the data content itself. By programming the transmitter's DSP to insert known data patterns, or “training sequences,” into the data stream, the receiver's DSP can learn how a given network configuration distorts the optical signal, based on the difference between the training sequences and the training sequences as received. With this knowledge, the receiver's DSP can customize its transfer functions to correct for the transmission impairments that have distorted the optical signal.
Ideally, the training sequences should be short in length but sufficiently complex so that the receiving DSP can quickly and efficiently develop a suitable transfer function that corrects for transmission distortions. To that end, network designers have developed known training sequences including constant-amplitude zero-autocorrelation (CAZAC) and Golay sequences. The paper Computational-Efficient and Modulation Format-Flexible Training-Aided Single-Carrier Digital Coherent Receiver (Tran et al., 2013), for example, describes the use of CAZAC and Golay training sequences, and is incorporated by reference.
FIG. 2 shows frames 1 through n of a dual polarization data stream that might be used with a PM-QAM transmission system as shown in FIG. 1. In this example, X-polarization data stream 200x includes data segments 204X1 through 204Xn, and Y-polarization data stream 200y includes data segments 204Y1 through 204Yn. Training sequences (TS) 2021 and 2023 are repeated in the X-polarization data stream and training sequences 2022 and 2024 are repeated in the Y-polarization data stream. TS 2021 is orthogonal to TS 2022 in the time domain, and TS 2023 is orthogonal to TS 2024 in the time domain. The data streams may also contain cyclic prefixes (not shown) to help eliminate interference between adjacent training sequences and data.
For training-aided frequency domain equalization (TA-FDE), the dual polarization data stream can be estimated using a transfer function T(ƒ) which is based on the received training sequences (RTSn) and the transmitted training sequences (TTSn) for two consecutive frames:
                              T          ⁡                      (            f            )                          =                                            [                                                                                                                  RTS                        1                                            ⁡                                              (                        f                        )                                                                                                                                                RTS                        3                                            ⁡                                              (                        f                        )                                                                                                                                                                                RTS                        2                                            ⁡                                              (                        f                        )                                                                                                                                                RTS                        4                                            ⁡                                              (                        f                        )                                                                                                        ]                        ⁡                          [                                                                                                                  TTS                        1                                            ⁡                                              (                        f                        )                                                                                                                                                TTS                        3                                            ⁡                                              (                        f                        )                                                                                                                                                                                TTS                        2                                            ⁡                                              (                        f                        )                                                                                                                                                TTS                        4                                            ⁡                                              (                        f                        )                                                                                                        ]                                            -            1                                              (                  eq          .                                          ⁢          1                )            
By repeating the calculation over the course of several frames, the transfer function can be further and further refined. For example, a newly calculated transfer function can be calculated from the previously calculated transfer function and the transfer function calculated from the current frame n as follows:TNEW(ƒ)=αTPREV(ƒ)+(1−α)Tn(ƒ)  (eq. 2)
where TNEW(ƒ) is the new transfer function as calculated over n frames, TPREV(ƒ) is the transfer function as calculated over n−1 frames, Tn(ƒ) is the transfer function calculated from the current frame n, and α is a forgetting factor with a typical value of 0.8˜0.9.
Techniques such as orthogonal frequency division multiplexing (OFDM), single carrier system with time domain equalization (SC-TDE), and single carrier system with frequency domain equalization (SC-FDE) have been used for data transmission. However, both OFDM and SC-TDE systems have disadvantages compared to SC-FDE systems. For example, OFDM systems experience sensitivity to non-linear impairments, phase noise, and frequency offset. Likewise, SC-TDE systems require analysis of many signal samples, referred to as “taps,” which complicates the processing requirements for extracting data from the data stream. In contrast, SC-FDE provides performance comparable to OFDM and SC-TDE without their respective disadvantages.
Although the conventional training sequences used perform comparably to SC-TDE when used with SC-FDE under ideal conditions, the performance drops sharply in real-world conditions, resulting in lower overall throughput. By way of example, chart 300 in FIG. 3 compares the required optical signal to noise ratio (rOSNR) for seven test simulations for a 34.4 GHz 16 QAM optical modulated signal using CAZAC training sequences with SC-TDE (13 taps) and a comparable signal using conventional training sequence aided SC-FDE. In the first three test cases, the transceivers are ideal (that is, there are no distortions caused by the transceiver components), while in the next four test cases, the transceivers use worst-case specifications of available optical and electrical transceiver components. As can be seen, for ideal transceivers (test cases 1-3) the performance of the SC-FDE configuration (shown as squares 301) is the same as the SC-TDE configuration (shown as triangles 302). However, where the transceivers are not ideal, the performance of the SC-FDE configuration is markedly inferior to that of the SC-TDE configuration. Thus a SC-FDE system can only achieve the same data throughput as SC-TDE over shorter transmission distances.
What is needed, therefore, are training sequences that can be used with SC-FDE and will lead to comparable or superior overall performance in comparison with SC-TDE.